Millimeter and sub-millimeter molecular-line observations of planetary nebula (PN) NGC 6537 (Red Spider) have been carried out using the Sub-Millimeter Telescope and the 12 m antenna of the Arizona Radio Observatory in the frequency range 86-692 GHz. CN, HCN, HNC, CCH, CS, SO, H 2 CO, HCO + and N 2 H + , along with the J = 3 → 2 and 6 → 5 lines of CO and those of several isotopologues, were detected toward the Red Spider, estimated to be ∼1600 yr old. This extremely high excitation PN evidently fosters a rich molecular environment. The presence of CS and SO suggest that sulfur may be sequestered in molecular form in such nebulae. A radiative transfer analysis of the CO and CS spectra indicate a kinetic temperature of T K ∼ 60-80 K and gas densities of n(H 2 ) ∼ 1-8 × 10 5 cm −3 in NGC 6537. Column densities of the molecules in the nebula and their fractional abundances relative to H 2 ranged from N tot ∼ 10 16 cm −2 and f ∼ 10 −4 for CO, to ∼7 × 10 11 cm −2 and f ∼ 8 × 10 −9 for the least abundant species, N 2 H + . For SO and CS, N tot ∼ 2 × 10 12 cm −2 and 10 13 cm −2 , respectively, with f ∼ 10 −7 and 2 × 10 −8 . It was also found that HCN/HNC ≈ 2. A low 12 C/ 13 C ratio of ∼4 was measured, indicative of hot-bottom burning. These results, coupled with past observations, suggest that molecular abundances in PNe are governed principally by the physical and chemical properties of the individual object and its progenitor star, rather than nebular age.